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On the Structure of Decision Diagram-Representable Mixed Integer Programs with Application to Unit Commitment

Hosseinali Salemi, Danial Davarnia

TL;DR

The present work addresses the question of which mixed-integer structures admit a DD representation by developing both necessary and sufficient conditions for a mixed- integer program to be DD-representable through identification of certain rectangular formations in the underlying sets.

Abstract

Over the past decade, decision diagrams (DDs) have been used to model and solve integer programming and combinatorial optimization problems. Despite successful performance of DDs in solving various discrete optimization problems, their extension to model mixed integer programs (MIPs) such as those appearing in energy applications has been lacking. More broadly, the question which problem structures admit a DD representation is still open in the DDs community. In this paper, we address this question by introducing a geometric decomposition framework based on rectangular formations that provides both necessary and sufficient conditions for a general MIP to be representable by DDs. As a special case, we show that any bounded mixed integer linear program admits a DD representation through a specialized Benders decomposition technique. The resulting DD encodes both integer and continuous variables, and therefore is amenable to the addition of feasibility and optimality cuts through refinement procedures. As an application for this framework, we develop a novel solution methodology for the unit commitment problem (UCP) in the wholesale electricity market. Computational experiments conducted on a stochastic variant of the UCP show a significant improvement of the solution time for the proposed method when compared to the outcome of modern solvers.

On the Structure of Decision Diagram-Representable Mixed Integer Programs with Application to Unit Commitment

TL;DR

The present work addresses the question of which mixed-integer structures admit a DD representation by developing both necessary and sufficient conditions for a mixed- integer program to be DD-representable through identification of certain rectangular formations in the underlying sets.

Abstract

Over the past decade, decision diagrams (DDs) have been used to model and solve integer programming and combinatorial optimization problems. Despite successful performance of DDs in solving various discrete optimization problems, their extension to model mixed integer programs (MIPs) such as those appearing in energy applications has been lacking. More broadly, the question which problem structures admit a DD representation is still open in the DDs community. In this paper, we address this question by introducing a geometric decomposition framework based on rectangular formations that provides both necessary and sufficient conditions for a general MIP to be representable by DDs. As a special case, we show that any bounded mixed integer linear program admits a DD representation through a specialized Benders decomposition technique. The resulting DD encodes both integer and continuous variables, and therefore is amenable to the addition of feasibility and optimality cuts through refinement procedures. As an application for this framework, we develop a novel solution methodology for the unit commitment problem (UCP) in the wholesale electricity market. Computational experiments conducted on a stochastic variant of the UCP show a significant improvement of the solution time for the proposed method when compared to the outcome of modern solvers.
Paper Structure (20 sections, 11 theorems, 15 equations, 9 figures, 3 tables, 3 algorithms)

This paper contains 20 sections, 11 theorems, 15 equations, 9 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background on Decision Diagrams
  3. Background on Unit Commitment Problem
  4. Contributions
  5. Decision Diagrams Representation
  6. Overview on Decision Diagrams
  7. Rectangular Decomposition
  8. Equivalence Class
  9. DD-Representable Sets
  10. Benders Decomposition
  11. Application to the Unit Commitment Problem
  12. Standard Formulations
  13. DD-BD: Master Problem Formulation
  14. DD-BD: Subproblem Formulation
  15. Computational Experiments
  16. ...and 5 more sections

Key Result

Theorem 1

Consider a compact set $\mathcal{P} \subseteq {\mathbb R}^n$, and select $I \subseteq N$. Assume that there exists a finite collection of compact sets $P_I^j$ for $j \in J$, where $J$ is an index set, such that Then, $\max\{f(\bm{x}) | \bm{x} \in \mathcal{P}\} = \max\{f(\bm{x}) | \bm{x} \in \bigcup_{j \in J} \mathcal{X}(P_I^j)\} = \max\{f(\bm{x}) | \bm{x} \in \bigcup_{j \in J} \bigcup_{k \in K_j}

Figures (9)

  • Figure 1: Set $\mathcal{P}$ of Example \ref{['ex:decompose']}
  • Figure 2: Set $\mathop{\rm proj}_{(x_1,x_2)}(\mathcal{P})$ of Example \ref{['ex:equivalence']}
  • Figure 3: The impact of equivalence class on DDs width. Numbers next to arcs represent the labels.
  • Figure 4: Different iterations of solving the master problem of Example \ref{['ex:MIP']}.
  • Figure 5: Illustration of an exact DD in Example \ref{['ex:DD-UCP']}
  • ...and 4 more figures

Theorems & Definitions (21)

  • Theorem 1
  • Example 1
  • Proposition 1
  • Lemma 1
  • Proposition 2
  • Definition 1
  • Corollary 1
  • Corollary 2
  • Example 2
  • Theorem 2
  • ...and 11 more